When Is a Topological Term Well Defined?

A proposed topological term is well defined only when its Lorentzian weight $e^{iS}$ assigns one phase to every admissible global field configuration. The answer must be independent of local representatives and auxiliary fillings, invariant under all allowed large transformations, and compatible with the declared boundary and tangential structure. A locally exact or metric-independent density has not yet passed those tests.

Two further questions come only after the phase exists. Are its fields fixed backgrounds or variables in the path integral? That choice separates a background response from a dynamical quantum theory. Does the resulting theory have an inverse under stacking? State spaces and stacking answer that independent question; a dynamical topological theory can itself be invertible. This page develops the tests on oriented manifolds and then applies one convention package to compact Abelian Chern–Simons theory, compact BF theory, and finite Dijkgraaf–Witten gauge theory in $2+1$ dimensions.

Required background. Local Potentials and Global Gauge Configurations supplies the patch data and compact-$U(1)$ flux lattice used below. Differential Forms, Integration, and Stokes’ Theorem supplies exterior differentiation, orientation, pullback, and the boundary formula.

Helpful background. De Rham Cohomology, Periods, Duality, and Intersection explains integral periods and why real differential forms miss torsion. Characteristic Classes and Chern–Weil Theory supplies the integral characteristic numbers used in the extension tests.

The exponentiated action is the global object

Start with the data that a local formula suppresses:

• the spacetime dimension, orientation, and any spin, framing, or other tangential structure;
• whether the manifold is closed, has a boundary, or contains prescribed defects;
• the global field type—ordinary form, connection on a bundle, differential cocycle, or finite cochain—and its charge or flux lattice;
• which fields are fixed backgrounds and which are integrated or summed; and
• the normalization of traces, characteristic classes, and the path-integral measure.

Only after those declarations does a displayed integral define a candidate action. In Lorentzian signature the invariant object is usually not a single-valued real number $S$, but its phase

$$\mathcal W[M;\Phi] := e^{iS[M;\Phi]}\in U(1),$$

where $\Phi$ denotes the complete global field configuration. Two local representatives are equivalent only if their actions differ by $2\pi\mathbb Z$. Thus a shift $S\mapsto S+2\pi n$ is invisible, while a shift by an arbitrary real number is not.

This distinction already separates three common statements. Let $\mathcal L_d$ be a top-degree density on a $d$-manifold.

1. Local exactness means that on every trivializing patch $U_i$ one can write $\mathcal L_d|_{U_i}=\mathrm dK_{d-1}^{(i)}$ with $K_{d-1}^{(i)}$ an admissible local expression in the fields and finitely many derivatives. An arbitrary chartwise primitive supplied by the Poincaré lemma is not enough. The admissible primitives need not agree on overlaps and need not assemble into a global form.

2. Metric independence means that the bulk metric variation vanishes. With

   $$T_{\mu\nu}^{\mathrm{bulk}} := -\frac{2}{\sqrt{\lvert g\rvert}} \frac{\delta S}{\delta g^{\mu\nu}},$$

   the test is $T_{\mu\nu}^{\mathrm{bulk}}=0$, after including hidden Hodge stars, index raising, regulators, and possible boundary terms.

3. Global well-definedness means that $\mathcal W[M;\Phi]$ descends to the full space of global configurations modulo every allowed gauge or coordinate transformation.

None implies the next. In particular, every $d$-form on a $d$-manifold is closed, so $\mathrm d\mathcal L_d=0$ is not a useful topological-term test. The Poincaré lemma also makes every smooth top form exact on a sufficiently small contractible chart, so restricting $K_{d-1}$ to the chosen algebra of field-local expressions is what makes local exactness meaningful. If $K_{d-1}$ is instead one globally defined smooth form and $M$ is closed, Stokes’ theorem gives $\int_M\mathrm dK_{d-1}=0$. Nonzero topological integrals therefore signal non-global primitives, boundaries, singularities, or global bundle data—not a failure of Stokes’ theorem. This local-versus- global distinction is explicit for characteristic forms and their Chern–Simons transgressions in Nakahara 2003, § 11.5.1, eqs. (11.100)–(11.102).

Five claims and the tests that separate them

The five phrases in the table are not mutually exclusive theories, nor are they a ladder of automatic implications. The first three concern the local and global definition of a functional. The last two additionally ask which fields are summed and whether the resulting quantum theory is invertible. Here a topological field theory (TQFT) means a dynamical quantum theory whose observables and state assignments satisfy metric-independent topological gluing; the later TQFT chapter develops the full operational definition. Formulae are kept in the surrounding prose so that every cell remains readable without mathematical typesetting.

Five distinct claims about a topological term; rows are tests, not mutually exclusive classes or automatic implications

Claim about the term	Decisive question	What must be specified or checked	What a pass does not imply
Local total derivative	Is the density the derivative of an admissible field-local expression on each trivializing patch?	Allowed local-form algebra and derivative order, patchwise primitive, overlap data, singular supports, and global exactness	A zero integral, a globally trivial phase, or invariance under large transformations
Metric-independent density	Does the bulk metric variation vanish after every hidden dependence is included?	Metric, orientation, regulator, gauge fixing, boundary terms, and stress-tensor convention	A globally defined action, a quantized coefficient, an invertible theory, or a TQFT
Globally defined exponentiated action	Is one phase assigned to each global configuration modulo all allowed transformations?	Bundles or cocycles, flux lattice, patch descent, large transformations, fillings, torsion, and boundary data	Invertibility, a dynamical theory, or the absence of boundary anomaly data
Quantized invertible response	Does the global test fix an allowed coefficient lattice or periodic domain, and with fields fixed does the response have a stacking inverse?	Allowed coefficient lattice or periodic identification, background family, local-counterterm convention, tangential structure, and one-dimensional state spaces	Intrinsic topological order or a theory obtained by summing the background
Dynamical TQFT	After integrating or summing fields, do observables and state spaces obey topological gluing?	Measure, gauge automorphisms, global sectors, state spaces, observables, boundary conditions, and framing or spin dependence	Either invertibility or noninvertibility; test stacking and state spaces separately

The table compresses seven independent checks that should be visible in a calculation:

1. vary the metric;
2. test local exactness without confusing it with global exactness;
3. descend through patches to a global configuration;
4. check every allowed large transformation;
5. compare all auxiliary extensions and derive the coefficient condition;
6. determine what changes at a boundary; and
7. declare fixed versus dynamical fields, then test invertibility and gluing.

Passing an early check cannot repair a failure at a later one. Conversely, a term can be globally defined without admitting a useful local density on one chart. Differential cocycles and state-sum definitions are designed precisely for that situation.

For a topological field theory, invertibility has a sharp state-space test: tensor-invertibility forces the state space on every closed codimension-one manifold to be a line, together with invertible amplitudes. A state space of dimension greater than one is therefore a decisive witness of noninvertibility Freed and Hopkins 2021, § 5.2, arXiv v6, printed p. 33, Open PDF.

Periods determine quantization or periodicity

An extension is often the quickest diagnostic. Suppose a candidate phase on $M^d$ is presented using a filling $Y^{d+1}$ with $\partial Y=M$:

$$\mathcal W[M;\Phi] = \exp\!\left( 2\pi i\,\kappa\int_Y\Omega_{d+1}(\widetilde\Phi) \right),$$

where $\widetilde\Phi$ extends $\Phi$. Two fillings $Y_1$ and $Y_2$ glue to the closed manifold $Z=Y_1\cup_M(-Y_2)$, so their ratio is

$$\frac{\mathcal W_{Y_1}}{\mathcal W_{Y_2}} = \exp\!\left( 2\pi i\,\kappa\int_Z\Omega_{d+1} \right).$$

The phase is filling-independent precisely when the exponent is an integer for every admissible closed $Z$ and every allowed extension of the fields. The same period test appears when a large gauge transformation is represented by a mapping torus. It can force a Chern–Simons, Wess–Zumino, or BF coefficient onto a discrete lattice.

Not every topological parameter is quantized this way. If a theory already has an intrinsic integer-valued charge $Q\in\mathbb Z$, then

$$e^{i\theta Q} = e^{i(\theta+2\pi)Q}$$

allows a continuous periodic angle $\theta\sim\theta+2\pi$. The distinction is whether the coefficient is needed to make a descended phase single-valued or instead weights already well-defined integer sectors.

The extension method also has a ceiling. A particular manifold or field configuration need not bound with all required structures. Real forms can miss torsion, and two extension prescriptions can differ by a bordism invariant. An intrinsic differential-cohomology construction replaces the filling by global cocycle data; the filling calculation remains a powerful necessary check, not a universal definition.

First application: one checklist for three topological theories

Work on a closed connected oriented three-manifold $M$ unless a boundary is explicitly introduced. A compact $U(1)$ connection has curvature $F$ with

$$\frac{1}{2\pi}\int_{\Sigma_2}F\in\mathbb Z$$

on every closed two-cycle. The minimal electric charge is one. All displayed actions use the Lorentzian phase $e^{iS}$; the corresponding Euclidean topological term carries the usual factor of $i$. These declarations make the three comparisons use the same flux and orientation conventions.

Compact Abelian Chern–Simons: the filling detects spin dependence

The familiar local expression is

$$S_{\mathrm{CS}}[a] = \frac{k}{4\pi}\int_M a\wedge\mathrm da.$$

It is metric independent and locally a transgression, but $a$ is not generally one global one-form. If $a$ and its bundle extend across an oriented four-manifold $X$ with $\partial X=M$, define the candidate phase by

$$e^{iS_{\mathrm{CS}}[a]} := \exp\!\left( \frac{ik}{4\pi}\int_X F\wedge F \right).$$

Changing the filling glues a closed four-manifold $Y$ and changes the action by

$$\Delta S_{\mathrm{CS}} = \frac{k}{4\pi}\int_Y F\wedge F = \pi k\,\bigl\langle c_1^2,[Y]\bigr\rangle.$$

On a general oriented four-manifold the integer $\langle c_1^2,[Y]\rangle$ can be odd. A one-component bosonic theory that does not require spin structure therefore needs even $k$. On a spin four-manifold, the Wu formula makes $c_1^2$ even, so every integer $k$ passes; odd $k$ defines a spin theory. In the lattice formulation, this is the distinction between an even integral bilinear form for ordinary Abelian Chern–Simons theory and an arbitrary integral form for spin Chern–Simons theory Belov and Moore 2005, §§ 1–2, arXiv v1, printed pp. 3–4 and 7–8, eqs. (1.1)–(1.3) and (2.2)–(2.7), Open PDF.

The filling is a diagnostic presentation. The appropriate differential-cohomology construction—together with its spin-dependent quadratic refinement when $k$ is odd—gives the intrinsic phase when an extension is unavailable. The filling test also does not settle the quantum theory’s framing dependence.

Field role now changes the answer without changing the local formula:

• With $a$ fixed as a background, the allowed phase is an invertible response; its stacking inverse has level $-k$.
• With $a$ integrated, the same expression defines a compact $U(1)_k$ Chern–Simons TQFT after its measure and framing data are supplied. For $\lvert k\rvert>1$ its torus state space has dimension $\lvert k\rvert$, so it is not invertible. For $k=\pm1$ the determinant formula gives a one-dimensional state space on every closed connected surface, so this particular obstruction disappears; full invertibility additionally requires the amplitude and global-data checks deferred to the specialized treatment. The case $k=0$ is degenerate rather than a nondegenerate Chern–Simons TQFT Belov and Moore 2005, § 5.3, arXiv v1, printed p. 26, prose following eq. (5.17), Open PDF.

Thus neither metric independence nor level quantization decides invertibility.

Compact BF: fixed cross-response versus a finite gauge TQFT

Let $a$ and $b$ be compact $U(1)$ connections. In three dimensions,

$$S_{\mathrm{BF}}[a,b] = \frac{N}{2\pi}\int_M b\wedge\mathrm da, \qquad N\in\mathbb Z_{>0}.$$

Integral flux and large gauge transformations change the action by $2\pi N$ times an integer, so $N$ must be integral in this normalization. Equivalently, it is the Abelian Chern–Simons theory with

$$K= \begin{pmatrix} 0 & N\\ N & 0 \end{pmatrix},$$

whose even diagonal makes the untwisted theory bosonic on oriented three-manifolds.

If both $a$ and $b$ are fixed backgrounds, the globally completed expression is an invertible cross-response phase. If both are integrated, it is the untwisted $\mathbb Z_N$ gauge TQFT. Its line operators

$$W_q(C)=\exp\!\left(iq\oint_Ca\right), \qquad V_p(C')=\exp\!\left(ip\oint_{C'}b\right)$$

with $p,q$ understood modulo $N$, obey on $S^3$ with the vacuum amplitude normalized to one

$$\bigl\langle W_q(C)V_p(C')\bigr\rangle_{S^3} = \exp\!\left( \frac{2\pi i}{N}\,qp\,\operatorname{Lk}(C,C') \right).$$

Reversing the orientation or linking convention complex-conjugates the phase. The theory has $N^2$ states on $T^2$, or equivalently $N^2$ simple electric–magnetic line sectors. It is therefore noninvertible for $N>1$. The compact action, its global completion and integer level, the Wilson operators, and their mutual phase are developed in Kapustin and Seiberg 2014, § 3, arXiv v2, printed pp. 9–13, especially eqs. (3.1)–(3.6) and (3.10), Open PDF. The state-space count follows independently from Belov and Moore 2005, § 5.3, arXiv v1, printed p. 26, prose following eq. (5.17), Open PDF.

The BF equations of motion enforce flatness locally. That fact alone does not derive the finite theory: compactness, the integral level, the global sector sum, and its normalization are essential.

Finite gauge theory: evaluation becomes a groupoid sum

Let $G$ be finite, let $P\to M$ be a flat principal $G$-bundle with classifying map $f_P:M\to BG$, and choose

$$[\omega]\in H^3(BG;\mathbb R/\mathbb Z) \cong H^3(BG;U(1)).$$

For fixed $P$, the Dijkgraaf–Witten weight is the unit phase

$$\mathcal W_\omega[M;P] = \exp\!\left( 2\pi i\, \bigl\langle f_P^*\omega,[M]\bigr\rangle \right).$$

Its cohomology class makes the result independent of cocycle representative. If $P$ is instead dynamical, the finite groupoid sum is

$$Z_{G,\omega}(M) = \sum_{[P]\in\pi_0\operatorname{Bun}^{\mathrm{flat}}_G(M)} \frac{\mathcal W_\omega[M;P]}{\lvert\operatorname{Aut}(P)\rvert}.$$

For a trivial input theory this defines the Dijkgraaf–Witten gauge theory. If the same operation is described as gauging a pre-existing global $G$ symmetry, its anomaly restricted to $G$ must first vanish or be cancelled. The bundle sum and its cocycle realization are the defining finite-gauge construction of Dijkgraaf and Witten 1990, §§ 6.2 and 6.4, printed pp. 415–416 and 420–423, especially eqs. (6.8)–(6.10) and (6.22)–(6.26), Open PDF. The automorphism denominator is the gauge-theory measure required by gluing, not an optional normalization. On a boundary, the classical action is a line rather than a number, and the quantum theory assigns state spaces to closed surfaces. These points are constructed explicitly in Freed and Quinn 1993, §§ 1–2, printed pp. 438–445, especially eqs. (1.1)–(1.2), (2.1), and (2.9), Open PDF.

For nontrivial $G$, the summed theory is generally noninvertible because it has multiple bundle and line sectors. For $G=\mathbb Z_N$ and trivial $[\omega]$, compact BF theory is its continuum presentation only after the global sectors, line operators, and groupoid normalization are matched. A local equation such as $\mathrm da=0$ is not enough to establish the equivalence.

The three cases now answer the principal question in one chain:

$$\text{local density} \longrightarrow \text{global phase} \longrightarrow \begin{cases} \text{fixed fields: response},\\ \text{summed fields: quantum theory}. \end{cases}$$

The first arrow is controlled by periods, large transformations, and boundary data. The second is a choice of field role. Invertibility is a further test on the resulting quantum theory. This fixed-background-versus-summed-field distinction is the operational definition of finite gauging in Gaiotto et al. 2015, §§ 1 and 6, arXiv v2, printed pp. 3–4 and 33, eqs. (1.3) and (6.1)–(6.2), Open PDF.

Boundaries require a completion

On a manifold with boundary, “topological” does not mean “nothing happens.” For a small Abelian gauge transformation $a\mapsto a+\mathrm d\lambda$,

$$\delta_\lambda S_{\mathrm{CS}} = \frac{k}{4\pi} \int_{\partial M}\lambda\,\mathrm da.$$

The bulk expression alone therefore fails to be a gauge-invariant number. One must choose a boundary condition, add a boundary counterterm, couple edge degrees of freedom, or interpret the bulk as an inflow theory whose state on $\partial M$ lies in a line. Different completions are different physical systems.

The same issue appears in BF and finite gauge theory. Gauge transformations that were redundant on a closed manifold can act on boundary data, and a finite-bundle weight can take values in a boundary line. Cutting and gluing then pairs states rather than multiplying ordinary numbers. A boundary variation is evidence that more data are needed; by itself it neither proves an inconsistency nor identifies a unique edge theory.

For the displayed BF representative, $b\mapsto b+\mathrm d\chi$ gives

$$\delta_\chi S_{\mathrm{BF}} = \frac{N}{2\pi}\int_{\partial M}\chi\,\mathrm da.$$

Writing the symmetric $K$-matrix representative moves part of this variation between $a$ and $b$ by a boundary term. The need for boundary data is invariant; its allocation among local representatives is not Kapustin and Seiberg 2014, § 5, arXiv v2, printed pp. 20–21, eqs. (5.1)–(5.3), Open PDF.

Tangential structure is equally consequential. Odd-level diagonal Abelian Chern–Simons theory needs spin structure. Quantization can retain a framing anomaly even when the classical density contains no metric. Replacing an oriented theory by a spin or framed theory changes its domain—it does not repair the original claim on the larger domain.

What the tests do not prove

The strongest reliable conclusion is the one attached to the last test that has actually passed.

• Locally exact does not mean globally trivial. The local primitives may fail to glue, and their overlap data can carry the entire topological term.
• Metric independent does not mean globally defined. A density can have zero bulk stress tensor and still fail a large-gauge or extension test.
• A quantized coefficient does not mean an invertible theory. Compact BF with integer $N>1$ is quantized but becomes a noninvertible finite gauge TQFT when its fields are summed.
• Dynamical does not mean noninvertible. Some fully dynamical topological theories are invertible; the state-space and stacking tests decide.
• No local propagating modes does not by itself define a TQFT. The measure, observables, global sectors, boundary conditions, and gluing law still have to exist and be metric independent.
• A filling formula is not automatically intrinsic. Nonbounding fields, torsion, and bordism invariants require a differential-cohomology or state-sum refinement.

These stops keep three distinct uses of the word “topological” from being silently identified: a property of a density, a property of an action phase, and a property of a quantum field theory.

Check your understanding

1. Why is closedness vacuous for a top-degree density?

Show why $\mathrm d\mathcal L_d=0$ does not distinguish a topological term on a $d$-manifold, and state the stronger local test.

Answer

The exterior derivative raises degree, so every $(d+1)$-form on a $d$-manifold vanishes. Hence every $d$-form is automatically closed. The stronger local statement is that on each trivializing patch $\mathcal L_d=\mathrm dK_{d-1}$ with $K_{d-1}$ an admissible local expression in the fields and finitely many derivatives—not merely an arbitrary Poincaré-lemma primitive. One then separately checks how those primitives glue and whether the exponentiated integral is global.

2. Recover the Abelian Chern–Simons level condition

For two fillings, use $\Delta S=\pi k\langle c_1^2,[Y]\rangle$. What follows on oriented and on spin four-manifolds?

Answer

An oriented four-manifold can have an integral class with odd self- intersection; $Y=\mathbb{CP}^2$ supplies the basic check. Then $e^{i\Delta S}=1$ for every allowed $Y$ requires even $k$. On a spin four-manifold the Wu formula makes every integral self-intersection even, so integer $k$ suffices. Odd $k$ therefore defines a spin-dependent theory, not an oriented non-spin theory.

3. Classify the two uses of compact BF theory

Keep $a$ and $b$ fixed in one experiment and integrate both in another. What changes?

Answer

With both fields fixed, the globally completed integer-level BF functional is a $U(1)$-valued cross-response and its inverse is the complex-conjugate phase. Integrating both compact fields sums global sectors and produces the $\mathbb Z_N$ gauge TQFT. For $N>1$ its $T^2$ state space has dimension $N^2$, so the dynamical theory is noninvertible even though each integrand is a phase.

4. Why does the finite-gauge measure contain automorphisms?

Explain why summing one unit for every representative flat bundle would be wrong.

Answer

Gauge-equivalent fields are the same object, and different bundles can have stabilizer groups of different sizes. The groupoid measure weights an isomorphism class by $1/\lvert\operatorname{Aut}(P)\rvert$. This is the finite analogue of dividing by gauge redundancy and is what makes cutting and gluing compatible.

5. Diagnose a boundary variation

If $\delta_\lambda S$ is a nonzero boundary integral, has the bulk theory been proved inconsistent?

Answer

No. The bulk expression is incomplete on that boundary domain. A consistent system may restrict the boundary fields, add an allowed counterterm, supply edge degrees of freedom whose variation cancels it, or treat the bulk as a relative/inflow theory. The calculation identifies the missing completion but does not choose one.

6. Separate quantization from periodicity

Why can a Chern–Simons level be discrete while a theta angle remains continuous?

Answer

The Chern–Simons coefficient is constrained so that different local or extension presentations give the same phase; the period test quantizes it. A theta angle multiplies an already integral charge $Q$, so any real $\theta$ defines a phase and only the identification $\theta\sim\theta+2\pi$ follows. Global form or fractional sectors can later refine that periodicity, but the two mechanisms remain different.

Choose the next specialized treatment

Continue according to the unresolved part of the test:

• Theta Terms, Periodicity, and Vacuum Sectors will develop integer sector weights, periodicity, and its global qualifications.
• Chern–Simons Actions and Level Quantization will develop non-Abelian normalization, spin and framing dependence, and boundary anomaly.
• Wess–Zumino and WZW Terms will develop extension independence and the boundary current-algebra handoff.
• BF Couplings and Discrete Topological Data will develop equations of motion, compactness, finite holonomy, and linking observables.
• Background Responses and Invertible Phases will develop stacking, local-counterterm equivalence, and anomaly inflow for fixed backgrounds.
• What Is a Topological Field Theory? will begin the state-space and gluing tests for dynamical theories.

Intrinsic differential-cohomology definitions and bordism classifications belong to the theorem-first mathematical treatment; no filling presentation on this page is meant to replace them.

References

• Belov, Dmitriy M., and Gregory W. Moore. “Classification of Abelian Spin Chern–Simons Theories.” arXiv:hep-th/0505235v1 [hep-th] (2005). Stable record.
• Dijkgraaf, Robbert, and Edward Witten. “Topological Gauge Theories and Group Cohomology.” Communications in Mathematical Physics 129, no. 2 (1990): 393–429. DOI. Open PDF.
• Freed, Daniel S., and Michael J. Hopkins. “Reflection Positivity and Invertible Topological Phases.” Geometry & Topology 25, no. 3 (2021): 1165–1330. DOI. Open PDF, arXiv v6.
• Freed, Daniel S., and Frank Quinn. “Chern–Simons Theory with Finite Gauge Group.” Communications in Mathematical Physics 156, no. 3 (1993): 435–472. DOI. Open published PDF. Current preprint, arXiv:hep-th/9111004v3.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 2015, no. 2 (2015): 172. DOI. Open PDF, arXiv v2.
• Kapustin, Anton, and Nathan Seiberg. “Coupling a QFT to a TQFT and Duality.” Journal of High Energy Physics 2014, no. 4 (2014): 001. DOI. Open PDF, arXiv v2.
• Nakahara, Mikio. Geometry, Topology and Physics. 2nd ed. Graduate Student Series in Physics. Bristol: Institute of Physics Publishing, 2003. ISBN 978-0-7503-0606-5. Publisher record.